In this paper, we consider the problem of computing the entire sequence of the maximum degree of minors of a block-structured symbolic matrix (a generic partitioned polynomial matrix) $A = (A_{αβ} x_{αβ} t^{d_{αβ}})$, where $A_{αβ}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$, $x_{αβ}$ is an indeterminate, and $d_{αβ}$ is an integer for $α= 1,2,\dots, μ$ and $β= 1,2,\dots,ν$, and $t$ is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum weight bipartite matching problem. The main result of this paper is a combinatorial $O(μν\min\{μ, ν\}^2)$-time algorithm for computing the entire sequence of the maximum degree of minors of a $(2 \times 2)$-type generic partitioned polynomial matrix of size $2μ\times 2ν$. We also present a minimax theorem, which can be used as a good characterization (NP $\cap$ co-NP characterization) for the computation of the maximum degree of minors of order $k$. Our results generalize the classical primal-dual algorithm (the Hungarian method) and minimax formula (Egerváry's theorem) for the maximum weight bipartite matching problem.